This question is essentially to come up with a solution for this problem in the case where there are $2^n$ coins. Let $C_n$ be the group $\{f : \mathbb Z / 2^n \mathbb Z \to \mathbb Z / 2 \mathbb Z\}$ equipped with the group operation pointwise addition. Let $R_n = \mathbb Z / 2 \mathbb Z \times ...

I am starting to take an interest in convex geometry and stumbled on the following theorem due to Pisier, the proof should be in https://link.springer.com/content/pdf/10.1007/BF01450929.pdf, although the theorem below is rephrased. Let $X$ be a subspace of $L_1(\Omega,\mu)$ of type $p$ for some $p>...

We consider the function $f:\mathbb R^2 \to \mathbb C.$ $$f(x,y) = 2i \sum_{k=0}^2 \omega^k \sin\left( \frac{x\bar \omega^k-y\omega^k}{2i} \right)$$ where $\omega= e^{2\pi i/3}.$ I computed using Mathematica the Taylor expansion at $x=y=0$ up to order $8$ $f(x,y) = \left(-\frac{y^5}{640}+O\left(y...

Let $d \colon \mathbb{R} \to \mathbb{R}$. Suppose that I have the following estimate for every $T>0$ $$|d(x)-d(y)|\leq C_T |x-y|+e^{-T}$$ where $C_T<0$ and $C_T \to +\infty$ as $T \to \infty$. Is it then possible to find a modulus of continuity (i.e. $\omega(0)=0$ and it is subadditive)?

Given the following matrix equation: $$AX=b$$ with: $A$ a (n,n) invertible matrix ; $X$ and $b$ vectors of size $n$ with all elements of $b$ positive or null, what is the condition on $A$ to have $X$ to have all his elements positive or null. I wonder if the response is so trivial... (i also fe...

I'm studying ODE. Over my math education career, I've occasionally encountered reference to homogeneous equations. Now, in ODE, I'm learning how to solve differential equations that are homogeneous. I've learned how to do this technique and how to recognize a homogeneous equation (and its order)....

We all knew, with $$\lim_{x\to 0}\frac{\sin x - x}{x(1-\cos x)}$$ we can use L'Hôpital's rule or Taylor series to eliminate undefined form. But without all tools, by only using high school knowledge, how can we evaluate this limit? It seems difficult to transform numerator, any idea? Thank you!

2a+b+c=0 6a+2b+c=0 I was wondering if there is a unique solution for a, b and c. I can't seem to come up with one with both equations equaling to 0. It would be greatly appreciated for some suggestions.

Consider an $n$-gon where we denote the points by $v_1, \dots, v_n$. If we allow each chord (internal edge of the $n$-gon) to have at most $k$ crossings, how many chords can we put into the $n$-gon (denoted as $c(n,k)$). The answer to this question is the density of outer-$k$-planar graphs (+ n f...

Let's say $a, b$ are zero divisors in a Ring, (i.e., there exist some $x,y \in R$ s.t. $ax=0, by=0$). I feel that $a$ is a zero divisor of $xy$ (as $axy=0y=0$), but is $b$ a zero divisor of $xy$? If I take a look at $bxy$, I know I can't commute $b$ and $x$, but can $b$ be a zero divisor of $(xy)$?

There are two local factories that produce microwaves. Each microwave produced at factory A is defective with probability $.05$, whereas each one produced at factory B is defective with probability $.01$. Suppose you purchase two microwaves that were produced at the same factory, which is equally...

Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$. Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$, a result which doesn't seem -understandably- to have any group theoretic proofs. K. Conrad ...

The following is given in my book as a proof of the remainder theorem: If $p(x)$ is the dividened, $q(x)$ is the quotient, $r(x)$ is the remainder and $(x-a)$ is the divisor, then: $$p(x) = q(x)(x-a) + r(x)$$ If $x=a$, $$\implies p(a) = q(a)(0) + r(a)$$ Hence, $$r(a) = p(a) \tag{i} $$ Therefore,...

Consider $t\ge 0$, $u=(u_1,u_2)^\top\in i\mathbb{R}^2$, $x=(x_1,x_2)^\top\in\mathbb{R}^2$ with $x_1\ge x_2^2$ and following complex number $$z:=-\frac{1}{2}\ln(1-2u_1t)+\frac{t\ u_2^2 + 2 \langle u,x\rangle}{2(1-2u_1t)}$$ How can I efficiently check, that $\Re(z)\le 0$ ?

I was going through a research paper based on elliptic divisibility sequences. In that paper the author has taken an elliptic curve over the rational field and a non-torsion point in it, for example the curve is given by $y^{2} = x^{3} + 80$ and the non-torsion point is $P=(4,12)$. By looking at ...

I'm working on a project analyzing how many research projects every EU region has been involved in in 2020. My lists show (1) the number of projects each region joined as coordinator, and (2) how many projects the region joined as participant. I want to generate a ratio between these two values t...

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable function. And let $b\in \mathbb{R}$ be a fixed number, consider the function $h:(-\infty,b]\rightarrow \mathbb{R} $ given by \begin{equation} h(x) = \min_{u \in [x,b]} f(u) \end{equation} I want to find $h'(x)$. Following the definitio...

Let $f:S^2\to S^2$ be a continuous with the property that it exists a nonempty,open $U\subseteq S^2$ with $f(x)=x$ for every $x\in U$ and $f(x)\notin U$ for $x\in S^2\setminus U$. Prove that $f$ is homotopic to $\operatorname{id}: S^2\to S^2$. My first idea was to take the homotopy $H: S^2\times ...

Let $C$ be a category, $X$ an object of $C$, and $i\colon A\to X$ and $j\colon B\to X$ monomorphisms (we say that $A$ and $B$ are subobjects of $X$, by abuse of language). Suppose these subobjects are disjoint. Does it follow that the union of $A$ and $B$ as subobjects is abstractly isomorphic to...